| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s + 6·13-s − 15-s − 6·17-s + 4·19-s + 25-s − 27-s − 2·29-s + 8·31-s − 4·33-s − 2·37-s − 6·39-s − 6·41-s − 12·43-s + 45-s − 8·47-s − 7·49-s + 6·51-s + 6·53-s + 4·55-s − 4·57-s − 12·59-s + 14·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s − 0.960·39-s − 0.937·41-s − 1.82·43-s + 0.149·45-s − 1.16·47-s − 49-s + 0.840·51-s + 0.824·53-s + 0.539·55-s − 0.529·57-s − 1.56·59-s + 1.79·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206444996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206444996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.83885673495833444158744201307, −11.34643695531064671651709069219, −10.28908580294693488189641431266, −9.234498555197333169762430563636, −8.372242663067272040741337380641, −6.71560882928791827542770443897, −6.23234003008114203357336027651, −4.86312932086210572949074191291, −3.57148112685573562185271725821, −1.48909737355432343830877670544,
1.48909737355432343830877670544, 3.57148112685573562185271725821, 4.86312932086210572949074191291, 6.23234003008114203357336027651, 6.71560882928791827542770443897, 8.372242663067272040741337380641, 9.234498555197333169762430563636, 10.28908580294693488189641431266, 11.34643695531064671651709069219, 11.83885673495833444158744201307
